derivative (ln(x)-1)/(ln^2(x))

解

導関数

\frac{ln ( x ) - 1}{ln ^{2} ( x )}

\frac{- ln ( x ) + 2}{x ln ^{3} ( x )}

解答ステップ

\frac{d}{d x} (\frac{ln ( x ) - 1}{ln ^{2} ( x )})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( ln ( x ) - 1 ) ln ^{2} ( x ) - \frac{d}{d x} ( ln ^{2} ( x ) ) ( ln ( x ) - 1 )}{( ln ^{2} ( x ) ) ^{2}}

\frac{d}{d x} (ln (x) - 1) = \frac{1}{x}

\frac{d}{d x} (ln^{2} (x)) = \frac{2 ln ( x )}{x}

= \frac{\frac{1}{x} ln ^{2} ( x ) - \frac{2 l n ( x )}{x} ( ln ( x ) - 1 )}{( ln ^{2} ( x ) ) ^{2}}

簡素化

\frac{\frac{1}{x} ln ^{2} ( x ) - \frac{2 l n ( x )}{x} ( ln ( x ) - 1 )}{( ln ^{2} ( x ) ) ^{2}} : \frac{- ln ( x ) + 2}{x ln ^{3} ( x )}

= \frac{- ln ( x ) + 2}{x ln ^{3} ( x )}